Homeomorphism theorems and a Green's formula for general elliptic boundary problems with nonnormal boundary conditions

Ya. A. Roitberg

Sbornik. Mathematics, Tome 12 (1970) no. 2, pp. 177-212

Résumé

Theorems on homeomorphisms for elliptic operators have been established by several mathematicians (Lions, Magenes, Schechter, Yu. M. Berezanskii, S. G. Krein, Ya. A. Roitberg and others; see RZhMat., 1962, 5B321; 1963, 5В268; 1964, 3B341, 12B286; 1965, 4B255; 1966, 3B321; 1968, 4B368, 11B309; 1969, 2B314.k). In their work it was always assumed that the boundary conditions are normal in the sense of Aronszajn–Milgram–Schechter. But in the present article the theorems on homeomorphisms are established without the assumption that the boundary expressions are normal; they may be pseudodifferential in the directions tangent to the boundary and differential in the normal directions. Essential use is made of the Green's formula developed for the case of the boundary conditions here in question. A study is also made of a problem formally adjoint to an elliptic problem with respect to Green's formula and the homeomorphism theorem is established for this problem as well. Bibliography: 36 titles.

Bibliographie

[1] J. L. Lions, E. Magenes, “Problemi ai limiti non omogenei. III, V”, Ann. Sci. Scuola Norm. Super Pisa, ser. III, 15:1–2 (1961), 39–101 ; 16:1 (1962), 1–44 ; “VI”, J. analyse math., 11 (1963), 166–188 | MR | Zbl | DOI | MR

[2] J. L. Lions, E. Magenes, Problemes aux limites non homogénes et applications, v. 1, Dunod, Paris, 1968

[3] Yu. M. Berezanskii, S. G. Krein, Ya. A. Roitberg, “Teorema o gomeomorfizmakh i lokalnoe povyshenie gladkosti vplot do granitsy reshenii ellipticheskikh uravnenii”, DAN SSSR, 148:4 (1963), 745–748 | MR | Zbl

[4] Ya. A. Roitberg, “Ellipticheskie zadachi s neodnorodnymi granichnymi usloviyami i lokalnoe povyshenie gladkosti vplot do granitsy obobschennykh reshenii”, DAN SSSR, 157:4 (1964), 798–804 | MR

[5] Ya. A. Roitberg, “Teorema o gomeomorfizmakh, osuschestvlyaemykh v $L_p$ ellipticheskimi operatorami, i lokalnoe povyshenie gladkosti obobschennykh reshenii”, Ukr. matem. zh., 17:5 (1966), 122–129 | MR

[6] Ya. A. Roitberg, “Teoremy o gomeomorfizmakh osuschestvlyaemykh ellipticheskimi operatorami”, DAN SSSR, 180:3 (1968), 542–545 | MR | Zbl

[7] Yu. M. Berezanskii, Ya. A. Roitberg, “Teorema o gomeomorfizmakh i funktsiya Grina dlya obschikh ellipticheskikh granichnykh zadach”, Ukr. matem. zh., 19:5 (1967), 3–32 | MR | Zbl

[8] E. Madzhenes, “Interpolyatsionnye prostranstva i uravneniya v chastnykh proizvodnykh”, Uspekhi matem. nauk, XXI:2(128) (1966), 169–218

[9] Yu. M. Berezanskii, Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Naukova dumka, Kiev, 1965 | MR

[10] M. Schechter, “On $L^p$ estimates and regularity. I, II”, Amer. J. Math., 85:1 (1963), 1–13 | DOI | MR | Zbl

[11] M. Schechter, “Negative norms and boundary problems”, Ann. Math., 72:3 (1960), 581–593 | DOI | MR | Zbl

[12] Ya. A. Roitberg, “Lokalnoe povyshenie gladkosti vplot do granitsy reshenii ellipticheskikh uravnenii”, Ukr. matem. zh., 15:4 (1963), 444–448 | MR | Zbl

[13] Yu. M. Berezanskii, Ya. A. Roitberg, “O gladkosti vplot do granitsy oblasti yadra rezolventy ellipticheskogo operatora”, Ukr. matem. zh., 15:2 (1963), 186–189

[14] Yu. M. Berezanskii, “O gladkosti .vplot do granitsy oblasti spektralnoi funktsii samosopryazhennogo differentsialnogo ellipticheskogo operatora”, DAN SSSR, 152:3 (1963), 511–514 | MR | Zbl

[15] Ya. A. Roitberg, “O razreshim osti obschikh granichnykh zadach dlya ellipticheskikh uravnenii pri nalichii stepennykh osobennostei v pravykh chastyakh”, Ukr. matem. zh., 20:3 (1968), 412–448 | MR

[16] S. G. Krein, A. S. Simonov, “Teorema o gomeomorfizmakh i kvazilineinye uravneniya”, DAN SSSR, 167:6 (1966), 1226–1229 | MR | Zbl

[17] S. G. Krein, “Povedenie reshenii ellipticheskikh zadach pri variatsii oblasti”, Studia Math., 31 (1968), 411–424 | MR | Zbl

[18] S. G. Krein, G. I. Laptev, “Abstraktnaya skhema rassmotreniya parabolicheskikh zadach v netsilindricheskikh oblastyakh”, Differents. uravneniya, V:8 (1969), 1458–1469 | MR

[19] Ya. A. Roitberg, “O granichnykh znacheniyakh obobschennykh reshenii ellipticheskikh uravnenii”, DAN SSSR, 188:1 (1969), 41–44 | MR | Zbl

[20] Ya. A. Roitberg, “Neodnorodnye ellipticheskie zadachi s razryvnymi koeffitsientami i lokalnoe povyshenie gladkosti obobschennykh reshenii vplot do granitsy oblasti i poverkhnosti razryva koeffitsientov”, Tretii Vsesoyuznyi simpoz. po difraktsii voln, referaty dokladov, Nauka, Moskva, 1964, 36–37

[21] G. Geymonat, “Su alcuni problemi ai limiti per i sistemi lineari ellittici secondo Petrowsky”, Matematiche, 20:2 (1965), 211–251 | MR

[22] Ya. A. Roitberg, Z. G. Sheftel, “Formula Grina i teorema o gomeomorfizmakh dlya ellipticheskikh sistem”, Uspekhi matem. nauk, XXII:5(137) (1967), 181–182 | MR

[23] Ya. A. Roitberg, Z. G. Sheftel, “Teorema o gomeomorfizmakh dlya ellipticheskikh sistem i ee prilozheniya”, Matem. sb., 78(120) (1969), 446–472 | MR | Zbl

[24] N. Aronszain, A. N. Milgram, “Differential operators on Riemannian manifolds”, Rend. Circ. Mat. Palermo, Ser. 2, 2 (1953), 1–61 | MR

[25] M. Schechter, “General boundary value problems for elliptic partial differential equations”, Comm. Pure and Appl. Math., 12:3 (1959), 457–486 ; Matematika, 4:5 (1960), 93–122 | DOI | MR | Zbl

[26] Yu. P. Krasovskii, “Svoistva funktsii Grina i obobschennye resheniya ellipticheskikh granichnykh zadach”, Izv. AN SSSR, seriya matem., 33 (1969), 109–137 | MR | Zbl

[27] M. I. Vishik, S. L. Sobolev, “Obschaya postanovka nekotorykh kraevykh zadach dlya ellipticheskikh differentsialnykh uravnenii v chastnykh proizvodnykh”, DAN SSSR, 111:3 (1956), 521–523 | Zbl

[28] S. G. Krein, Yu. I. Petunin, “Shkaly banakhovykh prostranstv”, Uspekhi matem. nauk, XXI:2(128) (1966), 89–168 | MR

[29] Ya. A. Roitberg, “Formula Grina i teorema o gomeomorfizmakh dlya obschikh ellipticheskikh granichnykh zadach s granichnymi usloviyami, ne yavlyayuschimisya normalnymi”, Ukr. matem. zh., 21:3 (1969), 406–413 | MR | Zbl

[30] Ya. A. Roitberg, “Teorema o gomeomorfizmakh dlya ellipticheskikh granichnykh zadach s granichnymi usloviyami, ne yavlyayuschimisya normalnymi”, DAN SSSR, 191:6 (1970), 1228–1231 | MR | Zbl

[31] M. S. Agranovich, “Ellipticheskie singulyarnye integro-differentsialnye operatory”, Uspekhi matem. nauk, XX:5(125) (1965), 3–120

[32] L. P. Volevich, “Razreshimost kraevykh zadach dlya obschikh ellipticheskikh sistem”, Matem. sb., 68(110) (1965), 373–416 | Zbl

[33] L. Khermander, Lineinye differentsialnye operatory s chastnymi proizvodnymi, Mir, Moskva, 1965 | MR

[34] R. T. Sili, “Integro-differentsialnye operatory na vektornykh rassloeniyakh”, Matematika, 11:2 (1967), 57–97

[35] M. Scheenter, “Various types of boundary conditions for elliptic equations”, Comm. Pure Appl. Math., 13:3 (1960), 407–425 | DOI | MR

[36] S. O. Ruschitskaya, “Teorema o gomeomorfizmakh dlya ellipticheskogo differentsialnogo vyrazheniya i psevdodifferentsialnykh granichnykh uslovii”, Ukr. matem. zh., 21:2 (1969), 268–273